Skip to the Main Content

Note:These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

April 20, 2007

Quantization and Cohomology (Week 21)

Posted by John Baez

This week in our course on Quantization and Cohomology we used Chen’s ‘smooth space’ technology to implement a new approach to Lagrangian mechanics, based on a smooth category equipped with an ‘action’ functor:

Week 21 (Apr. 17) - Any quotient of a smooth space becomes a
smooth space. The category of smooth spaces has pushouts.
The category of smooth spaces is cartesian closed. The path groupoid PXP X of a smooth space XX. The path groupoid is a smooth category. Smooth functors.
Theorem: a smooth functor S:PX→ℝS: P X \to \mathbb{R} is the same as a 1-form
on X.

In a bit more detail: we saw that any smooth space XX has a smooth groupoid of paths PXP X. The Lagrangian approach to classical mechanics involves a smooth groupoid CC where the objects are ‘configurations’ of our system and the morphisms are ‘processes’ or ‘paths’. The action should define a functor S:C→ℝS: C \to \mathbb{R}. So, it’s nice that in the special case when CC is a path groupoid, such functors turn out to be familiar entities! They’re just 1-forms on XX.

However, this isn’t quite general enough. What we really want is something that looks locally like a 1-form on XX, but not globally: a connection on a U(1)U(1) bundle over XX! This, after all, is what people use in geometric quantization — usually in the special case where XX is a symplectic manifold.

To get this answer, we’ll need to generalize from smooth functors to smooth anafunctors, as defined by Toby Bartels. A smooth anafunctor is something that’s locally isomorphic to a smooth functor!

Posted at April 20, 2007 7:39 PM UTC

TrackBack URL for this Entry: http://golem.ph.utexas.edu/cgi-bin/MT-3.0/dxy-tb.fcgi/1247

Read the post The n-Café Quantum ConjectureWeblog: The n-Category CaféExcerpt: Why it seems that quantum mechanics ought to be the de-refinement of a refined theory which lives in one categorical degree higher than usual.Tracked: June 8, 2007 11:39 AM